Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions

We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone op...

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Bibliographic Details
Published in:Mathematical programming Vol. 168; no. 1-2; pp. 645 - 672
Main Authors: Combettes, Patrick L., Eckstein, Jonathan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2018
Springer Nature B.V
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Convergence
Decomposition
Full Length Paper
Half spaces
Inclusions
Innovations
Linear operators
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Monotone inclusion
Primal-dual algorithm
65K05
90C25
Splitting algorithm
Duality
Monotone operator
49M27
Block-iterative algorithm
Asynchronous algorithm
47H05
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods. Flexible strategies are used to select the blocks of operators activated at each iteration. In addition, we allow lags in operator processing, permitting asynchronous implementation. The decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators, in order to construct a half-space containing the Kuhn–Tucker set associated with the system. The coordination phase of each iteration involves a projection onto this half-space. We present two related methods: the first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions. Neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators. Our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal-dual projective splitting methods.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-016-1044-0